## September 1, 2005

### Motivation

Today was the first class of the new semester. And this year, again, is my turn to teach String Theory. This year I decided to give a slightly extended version of my usual introductory lecture, explaining why one might want to study quantum gravity and, if so, why String Theory is the only plausible vehicle for studying it.

The arguments are old, but it occurred to me recently, that they are not as ubiquitously understood as they might be. So, as a public service, I decided to post a précis here.

I started off by recounting the tale of Howard Georgi, back in 1982, warning me off studying quantum gravity, as a waste of time. The point is that there’s no decoupling regime in which quantum “pure gravity” effects are important, while other particle interactions can be neglected. “Universality” in field theory — usually our friend — is, here, our enemy. Unless we know all particle physics interactions all the way from accessible energy up to the Planck scale, we can never hope to extract any quantitative predictions about quantum gravitational effects.

What about treating gravity classically? Maybe we don’t need to quantize it. Matter is certainly quantum mechanical. But, perhaps, we could just treat Einstein’s equation classically. Of course, if matter is quantum-mechanical, then $T_{\mu\nu}$ is an operator on Hilbert space, not a c-number. To make sense of the Einstein equation as a classical equation, we should presumably replace $T_{\mu\nu}$ by its expectation value, $\langle T_{\mu\nu}\rangle$.

The trouble is that $\langle T_{\mu\nu}\rangle$ depends nonlinearly on the state of the matter system. If we then solve the Einstein equation for the metric, and plug back in to contruct the time-evolution operator of the matter system, we discover that time-evolution is no longer given by a linear operator on Hilbert space. This is a disaster! People have tried to contruct nonlinear modifications of quantum mechanics. All such attempts have failed miserably.

Still, we manage, for the most part, to live happy, fulfilled lives while ignoring this colossal failure. As long as we can neglect the back-reaction of the matter system on the metric (i.e., as long as we do not attempt to solve the Einstein equation), we can perfectly well do quantum field theory in a fixed gravitational background. And we can happily do general relativity, provided we are willing to ignore the quantum mechanical nature of matter, i.e. to treat $T_{\mu\nu}$ as a classical c-number field.

That, however, won’t cut it if we want to study the quantum mechanics of blackholes, or the physics of the very early universe (for instance).

It’s often said that it is difficult to reconcile quantum mechanics (quantum field theory) and general relativity. That is wrong. We have what is, for many purposes, a perfectly good effective field theory description of quantum gravity. It is governed by a Lagrangian

(1)
$S = \int d^4 x \sqrt{-g} \left(M_{pl}^2 R + c_1 R^2 + c_2 R_{\mu\nu}^2 + \frac{c_3}{M_{pl}^2} R^3 +\dots +\mathcal{L}_{\text{matter}}\right)$

This is a theory with an infinite number of coupling constants (the $c_i$ and, all-importantly, the couplings in $\mathcal{L}_{\text{matter}}$). Nonetheless, at low energies, i.e., for $\varepsilon\equiv \frac{E^2}{M_{pl}^2}\ll 1$, we have a controllable expansion in powers of $\varepsilon$. To any finite order in that expansion, only a finite number of couplings contribute to the amplitude for some physical process. We have a finite number of experiments to do, to measure the values of those couplings. After that, everything else is a prediction.

In other words, as an effective field theory, gravity is no worse, nor better, than any other of the effective field theories we know and love.

The trouble is that all hell breaks loose for $\varepsilon\sim 1$. Then all of these infinite number of coupling become equally important, and we lose control, both computationally and conceptually.

An analogy with a more familiar case is helpful. Fermi theory (where we augment the original charged current interactions with neutral current ones, extend it beyond the first generation, …) is a perfectly adequate low-energy effective Lagrangian of the weak interactions. One has, schematically, a 4-Fermi interaction (a dimension-6 operator), $\frac{1}{M^2} \overline{\psi}\gamma_\mu P_L\psi \overline{\psi}\gamma^\mu P_L \psi$ plus the usual panoply of yet-higher-dimension operators, suppressed by higher powers of $M\sim 100$ GeV. For $\varepsilon=\frac{E^2}{M^2}\ll 1$, this is a perfectly valid effective field theory, which adequately described particle physics until 1983 (when the W and Z bosons were first produced experimentally). For $\varepsilon \sim 1$, it need to be replaced by some other theory, involving new degrees of freedom. Glashow, Weinberg and Salam won their Nobel prize in 1979 — i.e., well before the breakdown of Fermi theory — for proposing its successor.

So, one possibility for solving the problem of the breakdown of (1) is that new degrees of freedom appear at the Planck scale. But it’s not obvious what sort of degrees of freedom would come to our rescue here (not to mention the already uncomfortable fact that we don’t know what the degrees of freedom in $\mathcal{L}_{\text{matter}}$ should be, even below the Planck scale).

Another possibility is that the appearance of an infinite number of independent couplings is an illusion. Perhaps the physics of (1) is actually controlled by a UV fixed point. At the fixed point, the infinite number of couplings, rather than being independent, are actually functions of a small number of parameters, coordinatizing the fixed point set.

That’s a very attractive idea. Note that, unlike the previous case, where introducing new degrees of freedom might conceivably yield a theory which is weakly-coupled, the UV fixed-point theory is, almost certainly, going to be strongly-coupled, and so will need to be treated by nonperturbative methods.

There are various folks who purport to be attempting to “nonperturbatively” quantize general relativity. They are, secretly (whether they know it or not) pinning their hopes on the existence of a suitable UV fixed point. Otherwise, their theory has an infinite number of independently-adjustable couplings (and is, therefore, utterly unpredictive1). They may not have bothered to inquire about the number of independent couplings in their model but, barring a UV fixed point, you run into this problem, however studiously you avoid confronting it.

Moreover, if you’re pinning your hopes on a UV fixed point, Georgi’s objection now comes back to bite you. If you, somehow, managed to find a UV fixed point, and then you diddle with the matter content of your theory, you will end up totally missing the erstwhile UV fixed point. Adding matter to your theory cannot be an afterthought (as most of those hoping to “quantize GR” assume). You need to get your matter content right from the git-go. As Howard pointed out, that’s essentially impossible …

#### Escape

So, how does string theory beat these seemingly insurmountable odds?

It provide a unique, or nearly unique UV completion, not by having a fixed point, but by having a tower of higher-spin gauge symmetries that constrain the seemingly independent couplings of (1). In fact, they are so tightly constrained that there aren’t any continuously-adjustable coupling constants at all!

These higher spin gauge symmetries must be spontaneously-broken, and hence the corresponding higher spin “gauge fields” are massive (like the W’s and Z). But, even there, the existence of a consistent interacting theory of higher spin fields is only possible if they are infinite in number.

This is a funny trade-off. Rather than a finite number of (undetermined) UV degrees of freedom, with (undetermined) interactions, we have an infinite number. But their spectrum is completely constrained, and their interactions determined by the Ward identities associated to these broken higher spin gauge symmetries.

(One residuum of this is that, at very high energies, these higher spin gauge symmetries are effectively restored. The Coleman-Mandula Theorem then insures that the S-matrix is trivial. And, indeed, string scattering is very soft at high energies.)

#### Observables

Another conundrum of quantum gravity is what are the observables?

We know that proper observables must be gauge-invariant (i.e. diffeomorphism-invariant). In theories with asymptotic regions, you need to be a bit careful. You should only mod out by those gauge transformations which go to the identity at infinity. Those which act nontrivially at infinity are global symmetries of your theory (rather than gauge redundancies). Hence the existence of global charge in QED.

We only know of a few examples where the complete set of observables of quantum gravity have been determined. The answer depends very much on the asymptotics of our spacetime.

1. In asymptotically flat space, the group of residual diffeomorphisms is Poincaré. When disturbances are widely-separated, they are essentially non-interacting. So we start with some non-interacting degrees of freedom, classified by representations of Poincaré, in the far past; we end up with a similar set on non-interacting degrees of freedom in the far future. And we’re interested in the transformation from the “in” state to the “out” state. In other words, the observables of the theory are S-matrix elements.
2. In asymptotically anti-de Sitter space, the residual diffeomorphisms are the anti-de Sitter group, which is isomorphic to the conformal group in one lower dimension. Maldacena showed that the observables are the Green’s functions of a (conformal) quantum field theory on the boundary of our asymptotically AdS space.

These are very different-looking answers. No one knows, for instance, what the complete set of observables is, in the case of compact spatial topology. And you’re not going to be able to hazard a guess, based on these two examples.

#### Background Independence

I didn’t get to discuss this in class, and I think I’m not going to try to here. It’s been discussed to death at the String Coffee Table and on Cosmic Variance. No point in wasting more electrons rehashing those arguments.

#### Update: Discretized Theories

Since Wolfgang asks below, and since they seem to be the subject of rampant confusion, let me add a few, very general, remarks on discretized theories. Say we introduce a discretization, which thereby endows our theory with a fundamental length scale, $l$. At distance scales much longer than $l$, one expects to recover an effective continuum Lagrangian, of the general form (1), with coefficients which depend on the couplings of the discretized theory and on $l$.

Naïvely, one expects $M_{pl}^2\sim 1/l^2$. The “continuum limit” is obtained by taking $l\to 0$, while holding $M_{pl}$ fixed. That requires a fine-tuning of the couplings of the discretized theory. The required fine-tuning is awkward and, apparently, very difficult to achieve in practice. So some people have gotten the bright idea to stop trying, and simply leave $l$ fixed, rather than taking it to zero.

That would have made life much easier, had I not maliciously suppressed an important feature of (1). Namely, there’s a cosmological constant term, $\Delta S = \Lambda \int d^4 x \sqrt{-g}$ and, naïvely, $\Lambda \sim 1/l^4$. Even if we give up on give up on taking the continuum limt, so as to avoid the fine-tuning required to keep $M_{pl}^2\ll 1/l^2$, we still have to fine-tune in order to obtain $\Lambda\ll 1/l^4$.

Urs is right that LQG isn’t, strictly, a discretized model, though the use of the spin-network basis does introduce a fundamental length scale into the theory. It’s, more properly, a continuum theory, quantized in a Hamiltonian framework (albeit, a very, very unconventional one). The words I wrote above were geared to a Lagrangian formalism. It’s not hard to adapt them to a Hamiltonian one.

1 Except, of course, in the regime, $\varepsilon\ll 1$, where the effective field theory (1) can usefully be applied, because Universality comes to our rescue. If, by some miracle, we did know all of particle physics up to the Planck scale, that theory would be perfectly predictive within its realm of validity.

Posted by distler at September 1, 2005 2:37 AM

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### Re: Motivation

Hi Jacques, if I may be so bold, I might add a caveat to the discussion above.

It may be the case that gravity is asymptotically safe in Weinbergs sense. Namely while there may not exist a gaussian fixed UV point, it could be the case that there is a nongaussian fixed point.

In which case gravity could be nonperturbatively free of terrible divergences and we could consider (1) as a fundamental theory.

Something along the lines of this paper.

Feel free to debunk =)

Posted by: Haelfix on September 1, 2005 1:33 PM | Permalink | Reply to this

### UV Fixed Point

Yeah. I had hoped I was being clear.

I meant a nontrivial (non-Gaussian) UV fixed point. A Gaussian fixed point would be too much to hope for.

Posted by: Jacques Distler on September 1, 2005 1:55 PM | Permalink | PGP Sig | Reply to this
Read the post Approaches to Quantum Gravity
Weblog: Cosmic Variance
Excerpt: Over on Musings, Jacques has given a rather nice (technical) description of some of the motivations for looking beyond quantum field theory for an approach to formulating quantum gravity, and why the stringy approach is so promising. It supplements n...
Tracked: September 1, 2005 2:07 PM

### Re: Motivation

Jacques,

it seems another option not covered by your analysis is the idea that space-time is “discrete” and Lorentz symmetry is restored at larger distances only.
It seems to me that several “alternatives” (causal sets, dynamical triangulation, LQG (?) etc.) which do not attempt to find a true continuum limit fall into this category.

Posted by: Wolfgang on September 1, 2005 2:38 PM | Permalink | Reply to this

### Discretized Theories

They most certainly are covered in my discussion.

Discretizing spacetime is a fine way to regularize (and hence, make well-defined) the theory. Unless there is a UV fixed point, the discretized theory has an infinite number of independently-adjustable couplings, just as I described above. (Just because you choose not to write them down, doesn’t mean they are not there.)

At distance scales much longer than the discretization, Universality (modulo Georgi’s devastating caveat) comes to your rescue. At the scale of the discretization, the situation is exactly as I described it above.

Posted by: Jacques Distler on September 1, 2005 2:53 PM | Permalink | PGP Sig | Reply to this

### Re: Discretized Theories

> Unless there is a UV fixed point, the
> discretized theory has an infinite number
> just as I described above. (Just because
> you choose not to write them down, doesn’t
> mean they are not there.)

And what if we insist upon something like the nearest neighbor couplings? And let’s also assume that each vertex/edge/plaquette/cell only has finitely many states. I know it might seem a bit arbitrary, but isn’t that what microlocality is all about?

Posted by: Jason on September 1, 2005 3:35 PM | Permalink | Reply to this

### discretized?

I never understood why people say that LQG is an approach based on discretization. It’s not, as far as I can see.

The Hamiltonian approach to LQG writes down the Einstein-Hilbert action in the continuum and then tries to find some sort of operator analog for the ADM constraints derived from it. Nowhere is spacetime discretized.

What is true is that the most popular choice of basis for the kinematical Hilbert space which is used in LQG (the spin network basis), is sort of like a ‘delta function basis’. But that’s just a choice of basis and not even one of the physical Hilbert space.

It is also true that there are operators on that kinematical Hilbert space which are interpeted as surface observables and which take discrete eigenvalues in the spin network basis. But spin network states are not physical and hence even these discrete eigenvalues don’t show up for physical states. So, while LQG is clearly not based on a discretization, it is dubious that one can claim that a discrete spacetime somehow emerges from it.

One might argue that working on non-seperable (kinematical) Hilbert spaces corresponds sort of to a discretization of spacetime. I don’t think this is right, either. This step rather seems to correspond to putting a discrete topology on configuration space. It does not, however, change the content of config space (which consists of smooth spacetimes if that’s what one started with.)

Posted by: Urs Schreiber on September 1, 2005 3:31 PM | Permalink | Reply to this

### Re: discretized?

> The Hamiltonian approach to LQG writes
> down the Einstein-Hilbert action in the
> continuum

I agree, therefore I put the question mark next to LQG. But I guess one could use LQG to motivate discrete models a la Regge-Ponzano.

PS: Jacques I see that my 2nd reply was in the main thread and should probably be in the DISCRETIZED sub-thread. I am sorry, you blog is much more sophisticated than I am …

Posted by: Wolfgang on September 1, 2005 3:45 PM | Permalink | Reply to this

### Re: Motivation

What if, hypothetically speaking, nature turns out to have infinitely many adjustable parameters in reality (no UV fixed point), which however, do assume specific, but arbitrary values in the universe that we inhabit? Then it’s true that we lose predictivity at the Planck scale (in the sense that the only way we can figure out how nature is TRULY like at the Planck scale is to perform experiments at that scale, but that is possible in principle, if not in practice). Why do we have to rule out any theory which is “nonpredictive” in the above sense? What is the motivation for the widespread prejudice that the fewer adjustable free parameters a model has, the more likely it is to describe nature? Sure, it may be harder to calculate in practice (meaning impossible in practice) when we have infinitely many parameters of order one that we need to take into account. but in principle, the universe can still obey the laws set out by that theory in question. The universe isn’t obligated to be calculable by humans…

Posted by: Jason on September 1, 2005 3:03 PM | Permalink | Reply to this

### Unpredictive

Why do we have to rule out any theory which is “nonpredictive” in the above sense? What is the motivation for the widespread prejudice that the fewer adjustable free parameters a model has, the more likely it is to describe nature?

Careful, Jason. A certain self-anointed String Theory gadfly might hear you.

Posted by: Jacques Distler on September 2, 2005 10:37 AM | Permalink | PGP Sig | Reply to this

### Re: Motivation

> the discretized theory has an infinite
> couplings
I fail to see how this would be,
if you allow for the number of “lattice cells” to be large but finite (no true continuum limit).
Of course, if you attempt the usual a -> 0 limit you are back at the UV fixed point scenario.

Posted by: Wolfgang on September 1, 2005 3:04 PM | Permalink | Reply to this

### Re: Motivation

I would like to make one more attempt to clarify what I mean (but perhaps I am just to stubborn to understand the argument).
If some discrete theory uses a finite lattice spacing, e.g. a = 100L where L is the Planck length, then one would be in the region where epsilon (from your equ 1) is small, yet there would be enough room for a quasi-continuum limit at larger distances.

Posted by: Wolfgang on September 1, 2005 6:56 PM | Permalink | Reply to this

### Whither the cutoff?

I’m not sure we disagree on the physics. Perhaps it’s a matter of objectives.

1. We agree that we can’t take the cutoff $l\lt l_p$, because, absent some new input (new degrees of freedom, a UV fixed point, …), we don’t know what continuum theory replaces (1) at length scales shorter than $l_p$.
2. We agree that taking $l\sim l_p$ simply codifies our ignorance of the infinite number of couplings in (1), at the point where they are all equally important.
3. We could take the cutoff $l \gg l_p$, where the continuum effective Lagrangian works well. But that just amounts to saying that we refuse to entertain questions about the interesting regime, $\varepsilon\sim 1$. By definition, that regime simply isn’t in our theory.

Well, yeah, we could do that. But I think people would justifiably call that begging the question.

Posted by: Jacques Distler on September 2, 2005 12:06 AM | Permalink | PGP Sig | Reply to this

### Purely theoretical models

Besides, theorists aren’t obligated to restrict themselves to studying models which describe reality. In fact, a good number of the models that theorists work on (in the context of QFT, statistical mechanics, CFT, quantum gravity, string theory, etc.) are models which don’t describe anything that we know of in reality. So why do the theorists continue to work on such models? To obtain insights from them which might be generalizable to other models in general or simply to study a theoretical model for its own sake…

Let’s say a theorist comes up with a model of quantum gravity with a UV fixed point or maybe he or she fixes most of the infinitely many free parameters to some value (most likely zero. You might object that zero might not be stable under the renormalization group, but we might make some discrete cutoff and insist that they are zero for the discrete model). Of course, the probability that this particular model actually describes our universe is practically nil, especially if he or she neglects matter. But that model still exists and can be studied DESPITE the fact that it is not a theory of our universe. As you have repeatedly pointed out, adding matter will change the nature of the UV fixed points or will introduce many more parameters, but that theorist can still study various models with the primary goal of gaining qualitative insights. And by studying many such models, that theorist might get a feel (which may or may not be right) of what to expect from a generic theory of quantum gravity.

Posted by: Jason on September 1, 2005 3:25 PM | Permalink | Reply to this

### Re: Motivation

Yes, the cosmological constant …
But getting the empirical fact (?) of a very small but non-vanishing c.c. right is a tough challenge for every approach.

Posted by: Wolfgang on September 1, 2005 10:11 PM | Permalink | Reply to this

### Re: Motivation

What follows is something I should probably understand better than I do….

My confusion arises because Equation (1) is a Wilsonian effective action.
So if I want to calculate correlation functions of the fields
then I need to quantize this theory, and then
consider all loops generated, right?

On the other hand, in string theory imposing scale-invariance of an appropriate sigma
model also yields an effective action for the target space fields.
But, I *think* this latter case is
more analogous to a quantum effective action, in the sense that one
doesn’t usually go on and quantize this
effective action; I think people treat it as a set of classical equations of
motion for the expectation values of the target space fields. Is that
right?

So is the effective action usually derived in string theory really a
different beast to Equation (1)? If so, is it possible to say explicitly what
string theory constrains the coupling
constants in Equation (1) to be?

### Re: Motivation

My confusion arises because Equation (1) is a Wilsonian effective action. So if I want to calculate correlation functions of the fields then I need to quantize this theory, and then consider all loops generated, right?

Basically.

I meant it as an effective field theory action, à la Weinberg (as popularized, say, in Georgi’s Weak Interactions book). That is, at a cutoff scale, $\Lambda$, we integrate out all fields with masses $m\gt \Lambda$, keeping only the light fields, with $m\lt\Lambda$. (Wilson would keep all the fields, while integrating out their short-wavelength degrees of freedom. This is harder to implement in a covariant fashion.)

My discussion of how String Theory evades Georgi’s objection was based on String Field Theory. That’s not an entirely satisfactory approach, so take my discussion of String Theory, above, to be somewhat heuristic.

The main message, I hope, shines through the somewhat hand-wavy presentation. What saves us is the appearance of new symmetry principles and (a tower of) new degrees of freedom, which lead to a sort of “universal” behaviour in the UV.

Posted by: Jacques Distler on September 2, 2005 8:25 AM | Permalink | PGP Sig | Reply to this

### Re: Motivation

Jacques,

thanks for the answer. I think the conclusion is that string theory should uniquely fix the couplings of the low-energy action in Equation (1), but that it’s not necessarily easy to say what they are. Is that right? How in principle can one derive this kind of action from SFT?

Posted by: boreds on September 2, 2005 9:06 AM | Permalink | Reply to this

### Re: Motivation

“Not easy” is a bit of an understatement. :-)

In principle, one would take the String Field Theory action, integrate out all but a finite number of component fields (those whose masses are below the cutoff), and obtain the effective field theory for the “light” fields.

Unfortunately, for Closed String Field Theory, we — at best — know how to construct it perturbatively about some background. While the asymptotic spectrum is universal, the spectrum of light fields depends on what vacuum we choose.

This latter feature is not unprecedented. It happens all the time in any sufficiently complicated field theory (which has multiple vacua). The spectrum of light fields depends on the choice of vacuum.

But, the lack of a unified treatment of different closed string vacua in CSFT makes this computation rather painful to contemplate.

Still, there are nontrivial calculations that you can do, along these lines, if you have the motivation. One example is the computation by Dixon, Kaplunovsky and Louis of the 1-loop threshold corrections to the gauge couplings in the 4 dimensional effective theory.

Posted by: Jacques Distler on September 2, 2005 9:50 AM | Permalink | PGP Sig | Reply to this

### Re: Motivation

Something that occurs to me, which may be wrong - does it make sense to say that first-quantized string theory has nothing to say about quantum gravity, in the sense of saying something about Equation (1)?

Posted by: boreds on September 2, 2005 10:22 AM | Permalink | Reply to this

### Re: Motivation

Look at the computation of Dixon et al. It’s certainly done in first-quantized string theory. And it’s a prototype for the sort of computation one would have to do to construct the rest of (1).

Posted by: Jacques Distler on September 2, 2005 10:30 AM | Permalink | PGP Sig | Reply to this

### Re: Motivation

Thanks - I’ll try to digest that paper.

Posted by: boreds on September 2, 2005 10:53 AM | Permalink | Reply to this

### Re: Motivation

Jacques,

indeed we seem to agree on the physics (of discrete theories with finite cutoff a) but I have one more question about the limit where the length scale is very small compared to the Planck length. I assume that this would be (closely related to) the strong-coupling limit G -> infinity (or c -> 0 ). Martin Pilati found an exact solution for this case in 1982, without using a lattice or other such thing.
The reference is:
http://prola.aps.org/abstract/PRD/v26/i10/p2645_1
How does his result fit into your analysis ?

PS: Thank you for a great post and the patience to answer my questions.

Posted by: Wolfgang on September 2, 2005 7:26 AM | Permalink | Reply to this

### Pilati

Sorry, I’m not familiar with that paper (remember, Georgi told me to stop wasting my time thinking about quantum gravity in 1982!), so I’ll have to take a look before commenting.

[Yeah, I know, … that’s so un-blog-like!]

PS: Thank you for a great post and the patience to answer my questions.

I, in turn, would like to thank you and the others for the great comments. I think they really helped me to clarify the presentation here.

Next time I teach String Theory, this will be a really good lecture.

Posted by: Jacques Distler on September 2, 2005 8:53 AM | Permalink | PGP Sig | Reply to this

### Re: Motivation

>The trouble is that 〈T μν〉 depends nonlinearly on the state of the matter system. If we then solve the Einstein equation for the metric, and plug back in to contruct the time-evolution operator of the matter system, we discover that time-evolution is no longer given by a linear operator on Hilbert space. This is a disaster! People have tried to contruct nonlinear modifications of quantum mechanics. All such attempts have failed miserably.

I’d also like to understand what you mean here a bit better. I have no doubt this procedure is bad news, but what is the set of steps in principle that you’d go through here?

Are you supposed to know T_{\mu\nu} everywhere in space-time to begin with? And by the non-linear dependence on the state do you just mean having both bra and ket around T, or am I missing completely what you mean?

Posted by: boreds on September 2, 2005 10:01 AM | Permalink | Reply to this

### Re: Motivation

[If you want to use TeX in one of your comments, just turn on one of the “…itex to MathML” filters.]

Are you supposed to know $T_{\mu\nu}$ everywhere in space-time to begin with?

In the quantum theory of the “matter” system, it’s some definite operator.

And by the non-linear dependence on the state do you just mean having both bra and ket around $T$

Exactly.

So, at some fixed time, assuming we know the state, $|\Psi\rangle$, we know the RHS of the Einstein equation, $\langle\Psi|T_{\mu\nu}|\Psi\rangle$.

We’d now like to evolve this data forward in time.

The Einstein equation tells us how to evolve the metric forward in time. The metric, in turn, is used to construct the Hamiltonian of the matter system which — we were taught — tells us how to evolve the matter system forward in time.

Unfortunately, that’s no longer true. We don’t just get a linear operator, $e^{-i H t}$ governing the time-evolution of the matter system. The “Hamiltonian” we would obtain by following our noses, in this way, is a functional of the state, $|\Psi\rangle$.

At this point, I have to throw up my hands. If we attempt to proceed, we have to throw out one of the most cherished principles of quantum mechanics (linearity).

Posted by: Jacques Distler on September 2, 2005 10:24 AM | Permalink | PGP Sig | Reply to this

### Re: Motivation

I see, surprisingly that’s what I thought you meant. Just to check:

On some Cauchy surface you know: $|\psi(t_0)\rangle$ and an initial $g_{\mu\nu}(t_0)$ and these are enough to write down $\langle T_{\mu\nu}\rangle$.

This via Einstein gives $g_{\mu\nu}(t_0+\delta t)$

which gives $H=T_{00}(t_0+\delta t)$

which via Schrodinger (in principle*) gives $|\psi(t_0+\delta t)\rangle$

which gives $\langle T_{\mu\nu}\rangle$ at $t_0+\delta t$

and so on. Only the nonlinearity at * makes it truly horrible to do.

So is there a heuristic way to see how string theory sorts out this mess?

Posted by: boreds on September 2, 2005 11:21 AM | Permalink | Reply to this

### Re: Motivation

(I presume it’s because one is effectively removing the $\langle \rangle$ from $T_{\mu\nu}$)

Posted by: boreds on September 2, 2005 11:30 AM | Permalink | Reply to this

### Re: Motivation

So is there a heuristic way to see how string theory sorts out this mess?

String Theory treats the metric as a quantum field, on an equal footing as everything else.

This passage was a criticism of the idea to treat the metric classically, while keeping “matter” quantum-mechanical. In other words,it’s an argument for why we “need” to quantize gravity.

Posted by: Jacques Distler on September 2, 2005 11:33 AM | Permalink | PGP Sig | Reply to this

### Re: Motivation

Right, that’s pretty much what I meant above. Thanks Jacques

Posted by: boreds on September 2, 2005 11:42 AM | Permalink | Reply to this
Weblog: Musings
Excerpt: The last best hope of quantum gravity: a précis of the introductory lecture in my String Theory class.
Tracked: September 4, 2006 11:31 AM
Weblog: Musings
Excerpt: When "exact" doesn't really mean exact.
Tracked: September 4, 2006 11:38 AM
Weblog: Musings
Excerpt: Some thoughts on Causal Dynamical Triangulation models.
Tracked: September 4, 2006 11:38 AM
Read the post The LQG Landscape
Weblog: Musings
Excerpt: Smolin on coupling field theories to LQG.
Tracked: September 4, 2006 11:40 AM
Read the post Effective Field Theory and Gravity
Weblog: Musings
Excerpt: A bound on the number of e-foldings of inflation.
Tracked: April 27, 2007 3:26 AM

### The Correct Quantum Gravity Theory

For the correct quantum gravity theory, see the below paper:

F. J. Tipler, “The structure of the world from pure numbers,” Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964. http://math.tulane.edu/~tipler/theoryofeverything.pdf Also released as “Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything,” arXiv:0704.3276, April 24, 2007. http://arxiv.org/abs/0704.3276

The above paper demonstrates that the correct quantum gravity theory has existed since 1962, first discovered by Richard Feynman in that year, and independently discovered by Steven Weinberg and Bryce DeWitt, among others. But because these physicists were looking for equations with a finite number of terms (i.e., derivatives no higher than second order), they abandoned this qualitatively unique quantum gravity theory since in order for it to be consistent it requires an arbitrarily higher number of terms. …

[The rest of this comment rapidly descends into gibberish. I’m not interested in hosting long discussions of people’s pet theories of quantum gravity. — the editor]

Posted by: James Redford on January 6, 2008 9:19 PM | Permalink | Reply to this